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We are presented with a graph, $G$, on $n$ vertices and $m$ edges whose edge set is unknown. Our goal is to learn the edges of $G$ with as few queries to an oracle as possible. When we submit a set $S$ of vertices to the oracle, it tells us whether or not $S$ induces at least one edge in $G$. This so-called OR-query model has been well studied, with Angluin and Chen giving an upper bound on the number of queries needed of $O(m \log n)$ for a general graph $G$ with $m$ edges.

When we allow ourselves to make *quantum* queries (we may query subsets in superposition), then we can achieve speedups over the best possible classical algorithms. In the case where $G$ has maximum degree $d$ and is $O(1)$-colorable, Montanaro and Shao presented an algorithm that learns the edges of $G$ in at most $\tilde{O}( d^2 m^{3/4} )$ quantum queries. This gives an upper bound of $\tilde{O}( m^{3/4} )$ quantum queries when $G$ is a matching or a Hamiltonian cycle, which is significantly larger than the lower bound of $\Omega( \sqrt{m} )$ queries given by Ambainis and Montanaro.

We improve on the work of Montanaro and Shao in the case where $G$ has bounded degree. In particular, we present a randomized algorithm that, with high probability, learns cycles and matchings in $\tilde{O}( \sqrt{m} )$ quantum queries, matching the theoretical lower bound up to logarithmic factors.